Problem: If the sum of the squares of nonnegative real numbers $a,b,$ and $c$ is $39$, and $ab + bc + ca = 21$, then what is the sum of $a,b,$ and $c$?
Solution: Since $$(a+b+c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca = (39) + 2(21) = 81,$$ it follows that $a+b+c = \pm 9$. Since $a,b,c \ge 0$ we find $a+b+c=\boxed{9}$.